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Questions tagged [polynomials]

Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

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Variants of Eisenstein irreducibility

In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
3
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3answers
551 views

Polynomials and L^p(R)

As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
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2answers
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Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
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1answer
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Irreducible Polynomials in UFD and corresponding Quotient Field

Hello, "Let $D$ be a UFD and let $F$ be its quotient field. Further let $f$ be a primitive polynomial of positive degree in $D\left[x\right]$. From this it follows that that $f$ is irreducible in $D\...
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9answers
15k views

Irreducibility of polynomials in two variables

Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper. Does anyone know of similar results in the same vein? How about ...
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3answers
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How to factorize X^n - 1 in Z/pZ?

How do I factorize a polynomial $X^n - 1$ over $\mathbb{F}_p$? In particular I need to find factors of the polynomial $X^{3^3 - 1} - 1 = X^{26} - 1$ over $\mathbb{F}_3$.
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1answer
2k views

Finding unknown integer-valued polynomials using inequalities

I've come across this interesting inequalities problem recently, which seemed straight-forward at first glance but has proven interesting enough to ask about it here. Suppose you are given the ...
7
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3answers
2k views

Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by: The leaves 1 and $x$ for $x$ drawn from a class of variables; and Closed under the binary ...
31
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1answer
3k views

Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
51
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5answers
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Bizarre operation on polynomials

There I was, innocently doing some category theory, when up popped a totally outlandish operation on polynomials. It seems outlandish to me, anyway. I'd like to know if anyone has seen this ...
3
votes
1answer
725 views

Bernstein inequality for multivariate polynomial

Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max_{|z|=1} |Q(z)|$. So, are there ...
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3answers
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Sets that can be mapped onto R^n by a polynomial

The question was edited several times. Most recent version, suggested by Fedja: Does there exist an open set $U\subset \mathbb R^n$ (n>1) that contains balls ...
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4answers
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What primes divide the discriminant of a polynomial?

Given a monic polynomial $p(t) = t^n + ... + c_1 t + c_0$ with integer (or rational) coefficients and with roots $a_1, \dots a_n$, we can compute its discriminant, which is defined to be $\prod_{i< ...
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6answers
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English reference for a result of Kronecker?

Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference: Let $f$ be a monic polynomial with integer ...
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3answers
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Counting lattice points on an n-simplex

Imagine an n-simplex, the solution set for the expression: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where: $a_1$ through $a_n$ are positive bounded integers $x_1$ through $x_n$ are ...
246
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7answers
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Polynomial representing all nonnegative integers

Lagrange proved that every nonnegative integer is a sum of 4 squares. Gauss proved that every nonnegative integer is a sum of 3 triangular numbers. Is there a 2-variable polynomial $f(x,y) \in \...
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3answers
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When does a real polynomial have a pair of complex conjugate roots?

Suppose we have a polynomial function $f(z)=a_0+a_1z+a_2z^2+...+z^n$ with each $a_i$ between 0 and 1. Is there a method to determine if $f$ has a pair of complex conjugate roots? There are many ...
9
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1answer
693 views

what was Hilbert's geometric construction in his 17th problem?

Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
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12answers
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When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...
12
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4answers
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Is a polynomial with 1 very large coefficient irreducible?

I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum_{k=0}^{n-1} a_kx^k\...
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votes
3answers
277 views

Expressing field inclusions by polynomial equalities on coefficients

Let $A$ be the set of all quadruples $(a_0,a_1,a_2,a_3) \in {\mathbb Q}^4$ such that the polynomial $P=X^4+a_3X^3+a_2X^2+a_1X+a_0$ is irreducible and if $z$ is any root of $P$, then ${\mathbb Q}(z)$ ...
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1answer
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Tschirnhaus Transformation

Recently in my Intro to Proofs class, we've been talking about the fundamental theorem of algebra, which states that all polynomials of degree n always have n, not necessarily distinct, not ...
12
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2answers
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Irreducible polynomial over number field with roots in every completion?

Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions? ...
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5answers
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Two questions about discriminants of polynomials in Q[x]

Suppose $f \in \mathbb{Q}[x]$ is monic, with roots $\alpha_{1},\dots,\alpha_{n}$. Define the discriminant of $f$ to be the number $ \Delta = \Pi_{i<j} (\alpha_{i} - \alpha_{j})^2$. Let $D(f) = \...
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11answers
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Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
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5answers
2k views

Finding the new zeros of a “perturbed” polynomial

Given a univariate polynomial with real coefficients, p(x), with degree n, suppose we know all the zeros xj, and they are all real. Now suppose I perturb each of the coefficients pj (for j ≤ n) by ...
21
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2answers
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Least number of non-zero coefficients to describe a degree n polynomial

I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it. Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be ...
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4answers
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Are quotients of polynomial rings almost UFDs?

If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\...
70
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7answers
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Roots of truncations of $ e^x - 1$

During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has ...
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3answers
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No simple duplication formula for factorials?

Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads: Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2) On the ...
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7answers
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When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
8
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3answers
1k views

Polynomials that are sums of squares

Is any algorithm known for determining whether or not a multivariate polynomial with integer coefficients can be written as a sum of squares of such polynomials? By way of background, if we one ...
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3answers
557 views

Variant of binomial coefficients

I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc. First recall the following. If z is a ...
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1answer
972 views

Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function. There exists a nice product formula for the principal specializations: sλ...